I'm having this same issue on one of our iPhone 6 phones. Its weird because itll take multiple reboots but there are times it works perfectly normal. Where I can type, slide to unlock, browse the web etc.. Then all of a sudden the entire screen is just unresponsive. Any idea what this could be? Is it this so called touch disease people are saying? It seems like that issue is with iPhone 6 pluses ours is only a 6.
This is a common problem on a hardware "bend" on the top right common to the iPhone 6's.
People have opened them and put a sim card on the top right above or below the metal overlay there and had the problem disappear.
Look it up on YouTube and you'll learn about it and the common fixes people do.